Numerical methods for the solution of partial differential equations of fractional order

Lynch, V. E.; Carreras, B. A.; Del-Castillo-Negrete, Diego B; Ferreira-Mejias, K. M.; Hicks, H.R.

doi:10.1016/j.jcp.2003.07.008

Cited by 223 publications

(132 citation statements)

References 10 publications

(18 reference statements)

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“…An automatic quadrature method based on the Chebyshev polynomials was presented for approximating the Caputo derivative in [26]. Some other computational schemes, such as the L1, L2 and L2C schemes, etc., are also introduced [8,11,13,14,16,18,19,21,23,25,28,29,31].…”

Section: Introductionmentioning

confidence: 99%

Spectral approximations to the fractional integral and derivative

Zeng

Liu

2012

fcaa

151

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AbstractIn this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

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Section: Introductionmentioning

confidence: 99%

Spectral approximations to the fractional integral and derivative

Zeng

Liu

2012

fcaa

151

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“…We descritized the Riemann-Liouville fractional derivative by using the idea of Grunwald-Litnikove estimate as in [2] is as below, For simplification these coefficient will be denoted by,…”

Section: Mathematical Formulation Of the Model Problemmentioning

confidence: 99%

“…As we know that many schemes have been proposed to solve FDEs theoretically such as Laplace, Fourier transform method, Green function method, but most problems cannot be solved analytically and hence in this regard it is very remarkable to solve fractional differential equations numerically [1,2]. The whole conversation about the fractional differential equations are presented in [2].…”

Section: Introductionmentioning

confidence: 99%

“…The whole conversation about the fractional differential equations are presented in [2]. Numerous numerical method have been planned for the solution of time or spatial FDEs, for instance Finite difference method for time fractional diffusion equation which are presented in some studies, generated explicit and semi-implicit schemes for the solution of fractional order PDEs [3,4].…”

Section: Introductionmentioning

confidence: 99%

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Higher Order Compact Finite Difference Method for the Solution of 2-D Time Fractional Diffusion Equation

Usman¹,

Badshah²,

Ghaffar³

2018

Matrix sci. math.

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The main purpose of this study is to work on the solution of two-dimensional time fractional diffusion equation In this research work we apply the HOC scheme to approximate the second order space derivative. To obtain a discrete implicit scheme, Grunwald-Letnikov descritization is used in sense to approximate the RiemannLiouville time fractional derivative. The scheme thus obtained is based on block pentadiagonal matrix and each matrix has five-point stencil in order to reduce the computational cost we use AOS method. In AOS method, before taking the average of two solutions first we split the n-dimensional problems into a sum of n-one dimensional problem.

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“…Zhang and Han [30] proposed a quasi-wavelet method for time-dependent fractional partial differential equation. Lynch et al [20] proposed a numerical methods for the solution of partial differential equations of fractional order. In [8], Baskonus et al used active control method for fractional order economic system.…”

Section: Introductionmentioning

confidence: 99%

Numerical study of time-fractional hyperbolic partial differential equations

Arshed¹

2017

J. Math. Computer Sci.

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In this study, a numerical scheme is developed for the solution of time-fractional hyperbolic partial differential equation. In the proposed scheme, cubic B-spline collocation is used for space discretization and time discretization is obtained by using central difference formula. Caputo fractional derivative is used for time-fractional derivative. The stability and convergence of the developed scheme, have also been proved. The numerical examples support the theoretical results.

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Numerical methods for the solution of partial differential equations of fractional order

Cited by 223 publications

References 10 publications

Spectral approximations to the fractional integral and derivative

Spectral approximations to the fractional integral and derivative

Higher Order Compact Finite Difference Method for the Solution of 2-D Time Fractional Diffusion Equation

Numerical study of time-fractional hyperbolic partial differential equations

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